3.16 \(\int \frac{1}{(1+x^2)^{3/2}} \, dx\)

Optimal. Leaf size=11 \[ \frac{x}{\sqrt{x^2+1}} \]

[Out]

x/Sqrt[1 + x^2]

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Rubi [A]  time = 0.0011778, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {191} \[ \frac{x}{\sqrt{x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)^(-3/2),x]

[Out]

x/Sqrt[1 + x^2]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (1+x^2\right )^{3/2}} \, dx &=\frac{x}{\sqrt{1+x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0020271, size = 11, normalized size = 1. \[ \frac{x}{\sqrt{x^2+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)^(-3/2),x]

[Out]

x/Sqrt[1 + x^2]

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Maple [A]  time = 0.003, size = 10, normalized size = 0.9 \begin{align*}{x{\frac{1}{\sqrt{{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+1)^(3/2),x)

[Out]

x/(x^2+1)^(1/2)

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Maxima [A]  time = 0.948197, size = 12, normalized size = 1.09 \begin{align*} \frac{x}{\sqrt{x^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^(3/2),x, algorithm="maxima")

[Out]

x/sqrt(x^2 + 1)

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Fricas [B]  time = 0.420945, size = 53, normalized size = 4.82 \begin{align*} \frac{x^{2} + \sqrt{x^{2} + 1} x + 1}{x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^(3/2),x, algorithm="fricas")

[Out]

(x^2 + sqrt(x^2 + 1)*x + 1)/(x^2 + 1)

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Sympy [A]  time = 0.694002, size = 8, normalized size = 0.73 \begin{align*} \frac{x}{\sqrt{x^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+1)**(3/2),x)

[Out]

x/sqrt(x**2 + 1)

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Giac [A]  time = 1.07279, size = 12, normalized size = 1.09 \begin{align*} \frac{x}{\sqrt{x^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^(3/2),x, algorithm="giac")

[Out]

x/sqrt(x^2 + 1)